Calculus Examples

$\frac{lim_{x \to 5^{-}} 7 x^{2} - 1}{x^{2} - 25}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5^{-}} 7 x^{2} - lim_{x \to 5^{-}} 1}{x^{2} - 25}$

Step 1.2

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 lim_{x \to 5^{-}} x^{2} - lim_{x \to 5^{-}} 1}{x^{2} - 25}$

Step 1.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{7 {(lim_{x \to 5^{-}} x)}^{2} - lim_{x \to 5^{-}} 1}{x^{2} - 25}$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$\frac{7 {(lim_{x \to 5^{-}} x)}^{2} - 1 \cdot 1}{x^{2} - 25}$

Step 2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{7 \cdot 5^{2} - 1 \cdot 1}{x^{2} - 25}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $5$ to the power of $2$ .

$\frac{7 \cdot 25 - 1 \cdot 1}{x^{2} - 25}$

Step 3.1.2

Multiply $7$ by $25$ .

$\frac{175 - 1 \cdot 1}{x^{2} - 25}$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\frac{175 - 1}{x^{2} - 25}$

Step 3.1.4

Subtract $1$ from $175$ .

$\frac{174}{x^{2} - 25}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{174}{x^{2} - 5^{2}}$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{174}{(x + 5) (x - 5)}$